GENERALIZED DUNKL-SOBOLEV SPACES OF
EXPONENTIAL TYPE AND APPLICATIONS
Dunkl-Sobolev Spaces
HATEM MEJJAOLI
Department of Mathematics
Faculty of Sciences of Tunis
Campus-1060. Tunis, Tunisia.
EMail: hatem.mejjaoli@ipest.rnu.tn
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
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Contents
Received:
22 March, 2008
Accepted:
23 May, 2009
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Communicated by:
S.S. Dragomir
J
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2000 AMS Sub. Class.:
Primary 46F15. Secondary 46F12.
Key words:
Dunkl operators, Dunkl-Bessel-Laplace operator, Generalized Dunkl-Sobolev
spaces of exponential type, Pseudo differential-difference operators, Reproducing kernels.
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Abstract:
We study the Sobolev spaces of exponential type associated with the DunklBessel Laplace operator. Some properties including completeness and the imbedding theorem are proved. We next introduce a class of symbols of exponential
type and the associated pseudo-differential-difference operators, which naturally
act on the generalized Dunkl-Sobolev spaces of exponential type. Finally, using
the theory of reproducing kernels, some applications are given for these spaces.
Acknowledgement:
Thanks to the referee for his suggestions and comments.
Dedicatory:
Dedicated to Khalifa Trimèche.
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Contents
1
Introduction
3
2
Preliminaries
2.1 The Dunkl Operators . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Harmonic Analysis Associated with the Dunkl-Bessel Laplace Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
7
Dunkl-Sobolev Spaces
Hatem Mejjaoli
3
Structure Theorems on the Silva Space and its Dual
13
4
The Generalized Dunkl-Sobolev Spaces of Exponential Type
22
5
Applications
5.1 Pseudo-differential-difference operators of exponential type . . . . .
5.2 The Reproducing Kernels . . . . . . . . . . . . . . . . . . . . . . .
5.3 Extremal Function for the Generalized Heat Semigroup Transform .
30
30
34
38
vol. 10, iss. 2, art. 55, 2009
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1.
Introduction
The Sobolev space W s,p (Rd ) serves as a very useful tool in the theory of partial
differential equations, which is defined as follows
n
o
s,p
d
0
d
2 ps
p
d
W (R ) = u ∈ S (R ), (1 + ||ξ|| ) F(u) ∈ L (R ) .
In this paper we consider the Dunkl-Bessel Laplace operator 4k,β defined by
1
β≥− ,
2
where 4k is the Dunkl Laplacian on Rd , and Lβ is the Bessel operator on ]0, +∞[.
We introduce the generalized Dunkl-Sobolev space of exponential type WGs,p
(Rd+1
+ )
∗ ,k,β
s
2 p
by replacing (1 + ||ξ|| ) by an exponential weight function defined as follows
n
o
s,p
p
d+1
0
s||ξ||
d+1
WG∗ ,k,β (R+ ) = u ∈ G∗ , e FD,B (u) ∈ Lk,β (R+ ) ,
∀ x = (x0 , xd+1 ) ∈ Rd ×]0, +∞[,
4k,β = 4k,x0 + Lβ,xd+1 ,
where Lpk,β (Rd+1
+ ) it is the Lebesgue space associated with the Dunkl-Bessel trans0
form and G∗ is the topological dual of the Silva space. We investigate their properties
such as the imbedding theorems and the structure theorems. In fact, the imbedding
theorems mean that for s > 0, u ∈ WGs,p
(Rd+1
+ ) can be analytically continued to
∗ ,k,β
the set {z ∈ Cd+1 / | Im z| < s}. For the structure theorems we prove that for s > 0,
u ∈ WG−s,2
(Rd+1
+ ) can be represented as an infinite sum of fractional Dunkl-Bessel
∗ ,k,β
Laplace operators of square integrable functions g, in other words,
X sm
m
u=
(−4k,β ) 2 g.
m!
m∈N
We prove also that the generalized Dunkl-Sobolev spaces are stable by multiplication
of the functions of the Silva spaces. As applications on these spaces, we study
Dunkl-Sobolev Spaces
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vol. 10, iss. 2, art. 55, 2009
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the action for the class of pseudo differential-difference operators and we apply the
theory of reproducing kernels on these spaces. We note that special cases include:
the classical Sobolev spaces of exponential type, the Sobolev spaces of exponential
types associated with the Weinstein operator and the Sobolev spaces of exponential
type associated with the Dunkl operators.
We conclude this introduction with a summary of the contents of this paper. In
Section 2 we recall the harmonic analysis associated with the Dunkl-Bessel Laplace
operator which we need in the sequel. In Section 3 we consider the Silva space
G∗ and its dual G∗0 . We study the action of the Dunkl-Bessel transform on these
spaces. Next we prove two structure theorems for the space G∗0 . We define in Section
4 the generalized Dunkl-Sobolev spaces of exponential type WGs,p
(Rd+1
+ ) and we
∗ ,k,β
give their properties. In Section 5 we give two applications on these spaces. More
precisely, in the first application we introduce certain classes of symbols of exponential type and the associated pseudo-differential-difference operators of exponential
type. We show that these pseudo-differential-difference operators naturally act on
the generalized Sobolev spaces of exponential type. In the second, using the theory
of reproducing kernels, some applications are given for these spaces.
Dunkl-Sobolev Spaces
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
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2.
Preliminaries
In order to establish some basic and standard notations we briefly overview the theory of Dunkl operators and its relation to harmonic analysis. Main references are
[3, 4, 5, 8, 16, 17, 19, 20, 21].
2.1.
The Dunkl Operators
Dunkl-Sobolev Spaces
d
Let
space equipped with a scalar product h·, ·i and let ||x|| =
p R be the Euclidean
d
hx, xi. For α in R \{0}, let σα be the reflection in the hyperplane Hα ⊂ Rd
orthogonal to α, i.e. for x ∈ Rd ,
σα (x) = x − 2
hα, xi
α.
||α||2
A finite set R ⊂ Rd \{0} is called a root system if R ∩ R α = {α, −α} and σα R = R
for all α ∈ R. For a given root system R, reflections σα , α ∈ R, generate a finite
group
S W ⊂ O(d), called the reflection group associated with R. We fix a β ∈
Rd \ α∈R Hα and define a positive root system R+ = {α ∈ R | hα, βi > 0}. We
normalize each α ∈ R+ as hα, αi = 2. A function k : R −→ C on R is called a
multiplicity function if it is invariant under the action of W . We introduce the index
γ as
X
γ = γ(k) =
k(α).
α∈R+
Throughout this paper, we will assume that k(α) ≥ 0 for all α ∈ R. We denote by
ωk the weight function on Rd given by
Y
ωk (x) =
|hα, xi|2k(α) ,
α∈R+
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vol. 10, iss. 2, art. 55, 2009
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which is invariant and homogeneous of degree 2γ, and by ck the Mehta-type constant
defined by
Z
−1
2
ck =
exp(−||x|| )ωk (x) dx
.
Rd
We note that Etingof (cf. [6]) has given a derivation of the Mehta-type constant valid
for all finite reflection group.
The Dunkl operators Tj , j = 1, 2, . . . , d, on Rd associated with the positive root
system R+ and the multiplicity function k are given by
X
∂f
f (x) − f (σα (x))
Tj f (x) =
(x) +
k(α)αj
, f ∈ C 1 (Rd ).
∂xj
hα,
xi
α∈R
Dunkl-Sobolev Spaces
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
+
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We define the Dunkl-Laplace operator 4k on Rd for f ∈ C 2 (Rd ) by
4k f (x) =
d
X
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Tj2 f (x)
j=1
= 4f (x) + 2
X
α∈R+
k(α)
h∇f (x), αi f (x) − f (σα (x))
−
hα, xi
hα, xi2
,
where 4 and ∇ are the usual Euclidean Laplacian and nabla operators on Rd respectively. Then for each y ∈ Rd , the system
(
Tj u(x, y) = yj u(x, y), j = 1, . . . , d,
u(0, y) = 1
admits a unique analytic solution K(x, y), x ∈ Rd , called the Dunkl kernel. This
kernel has a holomorphic extension to Cd × Cd , (cf. [17] for the basic properties of
K).
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2.2.
Harmonic Analysis Associated with the Dunkl-Bessel Laplace Operator
In this subsection we collect some notations and results on the Dunkl-Bessel kernel,
the Dunkl-Bessel intertwining operator and its dual, the Dunkl-Bessel transform, and
the Dunkl-Bessel convolution (cf. [12]).
In the following we denote by
• Rd+1
= Rd × [0, +∞[.
+
Dunkl-Sobolev Spaces
0
• x = (x1 , . . . , xd , xd+1 ) = (x , xd+1 ) ∈
Rd+1
+ .
Hatem Mejjaoli
• C∗ (Rd+1 ) the space of continuous functions on Rd+1 , even with respect to the
last variable.
• C∗p (Rd+1 ) the space of functions of class C p on Rd+1 , even with respect to
the last variable.
• E∗ (Rd+1 ) the space of C ∞ -functions on Rd+1 , even with respect to the last
variable.
• S∗ (Rd+1 ) the Schwartz space of rapidly decreasing functions on Rd+1 , even
with respect to the last variable.
• D∗ (Rd+1 ) the space of C ∞ -functions on Rd+1 which are of compact support,
even with respect to the last variable.
0
d+1
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d+1
• S ∗ (R ) the space of temperate distributions on R , even with respect to
the last variable. It is the topological dual of S∗ (Rd+1 ).
We consider the Dunkl-Bessel Laplace operator 4k,β defined by
(2.1)
vol. 10, iss. 2, art. 55, 2009
∀ x = (x0 , xd+1 ) ∈ Rd ×]0, +∞[,
4k,β f (x) = 4k,x0 f (x0 , xd+1 ) + Lβ,xd+1 f (x0 , xd+1 ), f ∈ C∗2 (Rd+1 ),
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where 4k is the Dunkl-Laplace operator on Rd , and Lβ the Bessel operator on
]0, +∞[ given by
Lβ =
d2
2β + 1 d
+
,
2
dxd+1
xd+1 dxd+1
1
β>− .
2
The Dunkl-Bessel kernel Λ is given by
Λ(x, z) = K(ix0 , z 0 )jβ (xd+1 zd+1 ),
(2.2)
(x, z) ∈ Rd+1 × Cd+1 ,
where K(ix0 , z 0 ) is the Dunkl kernel and jβ (xd+1 zd+1 ) is the normalized Bessel function. The Dunkl-Bessel kernel satisfies the following properties:
i) For all z, t ∈ Cd+1 , we have
Dunkl-Sobolev Spaces
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
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(2.3) Λ(z, t) = Λ(t, z); Λ(z, 0) = 1 and Λ(λz, t) = Λ(z, λt), for all λ ∈ C.
ii) For all ν ∈ Nd+1 , x ∈ Rd+1 and z ∈ Cd+1 , we have
|Dzν Λ(x, z)| ≤ ||x|||ν| exp(||x|| || Im z||),
(2.4)
where Dzν =
∂ν
νd+1
ν
∂z1 1 ···∂zd+1
and |ν| = ν1 + · · · + νd+1 . In particular
|Λ(x, y)| ≤ 1,
(2.5)
for all x, y ∈ R
d+1
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The Dunkl-Bessel intertwining operator is the operator Rk,β defined on C∗ (Rd+1 )
by
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0
(2.6) Rk,β f (x , xd+1 )

Z xd+1
1

 √2Γ(β + 1) x−2β
(x2d+1 − t2 )β− 2 Vk f (x0 , t)dt,
d+1
1
πΓ(β + 2 )
0
=


0
f (x , 0),
xd+1 > 0,
xd+1 = 0,
where Vk is the Dunkl intertwining operator (cf. [16]).
Rk,β is a topological isomorphism from E∗ (Rd+1 ) onto itself satisfying the following transmutation relation
4k,β (Rk,β f ) = Rk,β (4d+1 f ),
(2.7)
where 4d+1 =
d+1
P
for all f ∈ E∗ (Rd+1 ),
∂j2 is the Laplacian on Rd+1 .
Dunkl-Sobolev Spaces
j=1
t
The dual of the Dunkl-Bessel intertwining operator Rk,β is the operator Rk,β
defined on D∗ (Rd+1 ) by: ∀y = (y 0 , yd+1 ) ∈ Rd × [0, ∞[,
Z ∞
1
2Γ(β + 1)
2
t
0
(s2 − yd+1
)β− 2 t Vk f (y 0 , s)sds,
(2.8)
Rk,β (f )(y , yd+1 ) = √
1
πΓ β + 2 yd+1
where t Vk is the dual Dunkl intertwining operator (cf. [20]).
t
Rk,β is a topological isomorphism from S∗ (Rd+1 ) onto itself satisfying the following transmutation relation
t
(2.9)
Rk,β (4k,β f ) = 4d+1 ( t Rk,β f ),
for all f ∈ S∗ (Rd+1 ).
d+1
We denote by Lpk,β (Rd+1
such that
+ ) the space of measurable functions on R+
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! p1
Z
|f (x)|p dµk,β (x) dx
||f ||Lp
=
||f ||L∞
= ess sup |f (x)| < +∞,
d+1
k,β (R+ )
d+1
k,β (R+ )
Rd+1
+
< +∞,
if
x∈Rd+1
+
where dµk,β is the measure on Rd+1
given by
+
0
dµk,β (x0 , xd+1 ) = ωk (x0 )x2β+1
d+1 dx dxd+1 .
1 ≤ p < +∞,
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The Dunkl-Bessel transform is given for f in L1k,β (Rd+1
+ ) by
(2.10)
Z
0
FD,B (f )(y , yd+1 ) =
Rd+1
+
f (x0 , xd+1 )Λ(−x, y)dµk,β (x),
for all y = (y 0 , yd+1 ) ∈ Rd+1
+ .
Some basic properties of this transform are the following:
Dunkl-Sobolev Spaces
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i) For f in L1k,β (Rd+1
+ ),
vol. 10, iss. 2, art. 55, 2009
||FD,B (f )||L∞
(2.11)
d+1
k,β (R+ )
≤ ||f ||L1
d+1
k,β (R+ )
.
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ii) For f in S∗ (R
(2.12)
d+1
) we have
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FD,B (4k,β f )(y) = −||y||2 FD,B (f )(y),
for all y ∈ Rd+1
+ .
iii) For all f ∈ S(Rd+1
∗ ), we have
(2.13)
FD,B (f )(y) = Fo ◦ t Rk,β (f )(y),
for all y ∈ Rd+1
+ ,
where Fo is the transform defined by: ∀ y ∈ Rd+1
+ ,
Z
0 0
(2.14) Fo (f )(y) =
f (x)e−ihy ,x i cos(xd+1 yd+1 )dx,
Rd+1
+
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f ∈ D∗ (Rd+1 ).
d+1
1
iv) For all f in L1k,β (Rd+1
+ ), if FD,B (f ) belongs to Lk,β (R+ ), then
Z
(2.15)
f (y) = mk,β
FD,B (f )(x)Λ(x, y)dµk,β (x), a.e.
Rd+1
+
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where
mk,β =
(2.16)
c2k
d
4γ+β+ 2 (Γ(β + 1))2
.
v) For f ∈ S∗ (Rd+1 ), if we define
FD,B (f )(y) = FD,B (f )(−y),
Dunkl-Sobolev Spaces
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then
vol. 10, iss. 2, art. 55, 2009
FD,B FD,B = FD,B FD = mk,β Id.
(2.17)
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Proposition 2.1.
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i) The Dunkl-Bessel transform FD,B is a topological isomorphism from S∗ (R
onto itself and for all f in S∗ (Rd+1 ),
Z
Z
2
(2.18)
|f (x)| dµk,β (x) = mk,β
|FD,B (f )(ξ)|2 dµk,β (ξ).
Rd+1
+
d+1
)
Rd+1
+
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1
2
ii) In particular, the renormalized Dunkl-Bessel transform f → mk,β FD,B (f ) can
be uniquely extended to an isometric isomorphism on L2k,β (Rd+1
+ ).
By using the Dunkl-Bessel kernel, we introduce a generalized translation and a
convolution structure. For a function f ∈ S∗ (Rd+1 ) and y ∈ Rd+1
the Dunkl-Bessel
+
translation τy f is defined by the following relation:
FD,B (τy f )(x) = Λ(x, y)FD,B (f )(x).
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If f ∈ E∗ (Rd+1 ) is radial with respect to the d first variables, i.e. f (x) = F (||x0 ||, xd+1 ),
then it follows that
p
(2.19) τy f (x) = Rk,β F
||x0 ||2 + ||y 0 ||2 + 2hy 0 , ·i,
q
2
2
xd+1 + yd+1 + 2yd+1 (x0 , xd+1 ).
Dunkl-Sobolev Spaces
By using the Dunkl-Bessel translation, we define the Dunkl-Bessel convolution product f ∗D,B g of functions f, g ∈ S∗ (Rd+1 ) as follows:
Z
(2.20)
f ∗D,B g(x) =
τx f (−y)g(y)dµk,β (y).
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vol. 10, iss. 2, art. 55, 2009
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Rd+1
+
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This convolution is commutative and associative and satisfies the following:
d+1
i) For all f, g ∈ S∗ (Rd+1
+ ), f ∗D,B g belongs to S∗ (R+ ) and
(2.21)
FD,B (f ∗D,B g)(y) = FD,B (f )(y)FD,B (g)(y).
kf ∗D,B gkLr
d+1
k,β (R+ )
≤ kf kLp
d+1
k,β (R+ )
kgkLq
d+1
k,β (R+ )
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ii) Let 1 ≤ p, q, r ≤ ∞ such that p1 + 1q − 1r = 1. If f ∈ Lpk,β (Rd+1
+ ) and g ∈
q
d+1
d+1
r
Lk,β (R+ ) is radial, then f ∗D,B g ∈ Lk,β (R+ ) and
(2.22)
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3.
Structure Theorems on the Silva Space and its Dual
Definition 3.1. We denote by G∗ or G∗ (Rd+1 ) the set of all functions ϕ in E∗ (Rd+1 )
such that for any h, p > 0
p||x|| µ
e
|∂ ϕ(x)|
Np,h (ϕ) = sup
h|µ| µ!
x∈Rd+1
µ∈Nd+1
Dunkl-Sobolev Spaces
is finite. The topology in G∗ is defined by the above seminorms.
Lemma 3.2. Let φ be in G∗ . Then for every h, p > 0
Np,h (ϕ) = sup
x∈Rd+1
m∈N
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vol. 10, iss. 2, art. 55, 2009
ep||x|| |4m
k,β ϕ(x)|
hm m!
!
.
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Proof. We proceed as in Proposition 5.1 of [13], and by a simple calculation we
obtain the result.
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Theorem 3.3. The transform FD,B is a topological isomorphism from G∗ onto itself.
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Proof. From the relations (2.8), (2.9) and Lemma 3.2 we see that t Rk,β is continuous
from G∗ onto itself. On the other hand, J. Chung et al. [1] have proved that the
classical Fourier transform is an isomorphism from G∗ onto itself. Thus from the
relation (2.13) we deduce that FD,B is continuous from G∗ onto itself. Finally since
G∗ is included in S∗ (Rd+1 ), and FD,B is an isomorphism from S∗ (Rd+1 ) onto itself,
by (2.17) we obtain the result.
We denote by G∗0 or G∗0 (Rd+1 ) the strong dual of the space G∗ .
Definition 3.4. The Dunkl-Bessel transform of a distribution S in G∗0 is defined by
hFD,B (S), ψi = hS, FD,B (ψ)i,
ψ ∈ G∗ .
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The result below follows immediately from Theorem 3.3.
Corollary 3.5. The transform FD,B is a topological isomorphism from G∗0 onto itself.
Let τ be in G∗0 . We define 4k,β τ , by
h4k,β τ, ψi = hτ, 4k,β ψi,
for all ψ ∈ G∗ .
This functional satisfies the following property
Dunkl-Sobolev Spaces
FD,B (4k,β τ ) = −||y||2 FD,B (τ ).
(3.1)
Definition 3.6. The generalized heat kernel Γk,β is given by
||x||2 +||y||2
2ck
x
y
−
4t
Γk,β (t, x, y) :=
e
Λ −i √ , √
;
d
2t 2t
Γ(β + 1)(4t)γ+β+ 2 +1
x, y ∈ Rd+1
+ , t > 0.
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The generalized heat kernel Γk,β has the following properties:
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Proposition 3.7. Let x, y in Rd+1
and t > 0. Then we have:
+
Z
i) Γk,β (t, x, y) =
exp(−t||ξ||2 )Λ(x, ξ)Λ(−y, ξ)dµk,β (ξ).
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+
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ii)
Rd+1
+
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Γk,β (t, x, y)dµk,β (x) = 1.
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iii) For fixed y in Rd+1
+ , the function u(x, t) := Γk,β (t, x, y) solves the generalized
heat equation:
4k,β u(x, t) =
∂
u(x, t)
∂t
on
Rd+1
+ ×]0, +∞[.
Definition 3.8. The generalized heat semigroup (Hk,β (t))t≥0 is the integral operator
given for f in L2k,β (Rd+1
+ ) by
 Z


Γk,β (t, x, y)f (y)dµk,β (y) if t > 0,
Rd+1
Hk,β (t)f (x) :=
+


f (x)
if t = 0.
From the properties of the generalized heat kernel we have
(
f ∗D,B pt (x) if t > 0,
(3.2)
Hk,β (t)f (x) :=
f (x)
if t = 0,
where
pt (y) =
2ck
γ+β+ d2 +1
Γ(β + 1)(4t)
e−
||y||2
4t
Dunkl-Sobolev Spaces
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vol. 10, iss. 2, art. 55, 2009
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.
Definition 3.9. A function f defined on Rd+1
+ is said to be of exponential type if there
are constants k, C > 0 such that for every x ∈ Rd+1
+
|f (x)| ≤ exp k||x||.
The following lemma will be useful later. For the details of the proof we refer to
Komatsu [9]:
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Lemma 3.10. For any L > 0 and ε > 0 there exist a function v ∈ D(R) and a
differential operator P ( dtd ) of infinite order such that
L
(3.3)
supp v ⊂ [0, ε], |v(t)| ≤ C exp −
, 0 < t < ∞;
t
(3.4)
P
d
dt
=
∞
X
ak
k=0
d
dt
P
(3.5)
k
d
dt
,
|ak | ≤ C1
Lk1
,
k!2
0 < L1 < L;
v(t) = δ + ω(t),
Dunkl-Sobolev Spaces
Hatem Mejjaoli
where ω ∈ D(R), supp ω ⊂ [ 2ε , ε].
vol. 10, iss. 2, art. 55, 2009
Here we note that P (4k,β ) is a local operator where 4k,β is a Dunkl-Bessel
Laplace operator.
Now we are in a position to state and prove one of the main theorems in this
section.
G∗0
Theorem 3.11. If u ∈ then there exists a differential operator
some C > 0 and L > 0,
X
n
∞
d
d
Ln
P
=
an
, |an | ≤ C 2 ,
dt
dt
n!
n=0
P ( dtd )
such that for
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Page 16 of 44
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and there are a continuous function g of exponential type and an entire function h of
exponential type in Rd+1
such that
+
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u = P (4k,β )g(x) + h(x).
Close
(3.6)
Proof. Let U (x, t) = huy , Γk,β (t, x, y)i. Since pt belongs to G∗ for each t > 0,
U (x, t) is well defined and real analytic in (Rd+1
+ )x for each t > 0.
Furthermore, U (x, t) satisfies
(3.7)
(∂t − 4k,β )U (x, t) = 0
for
(x, t) ∈ Rd+1
+ ×]0, ∞[.
Here u ∈ G∗0 means that for some k > 0 and h > 0
|hu, φi| ≤ C sup
(3.8)
x,α
|∂ α φ(x)| exp k||x||
,
h|α| α!
φ ∈ G∗ .
By Cauchy’s inequality and relations (2.4) and (3.8) we obtain, for t > 0
1
0
0
|U (x, t)| ≤ C exp k ||x|| + t +
t
0
0
for some C > 0 and k > 0. If we restrict this inequality on the strip 0 < t < ε then
it follows that
1
|U (x, t)| ≤ C exp k ||x|| +
, 0<t<ε
t
for some constants C > 0 and k > 0. Now let
Z
G(y, t) =
Γk,β (t, x, y)φ(x)dµk,β (x),
Rd+1
+
G(·, t) → φ
φ ∈ G∗ .
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Page 17 of 44
in
G∗ as t → 0+
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and
Close
Z
(3.10)
Rd+1
+
U (x, t)φ(x)dµk,β (x) = huy , G(y, t)i.
Then it follows from (3.9) and (3.10) that
(3.11)
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
Contents
Moreover, we can easily see that
(3.9)
Dunkl-Sobolev Spaces
lim U (x, t) = u
t→0+
in G∗0 .
Now choose functions u, w and a differential operator of infinite order as in Lemma
3.10. Let
Z ∞
(3.12)
Ũ (x, t) =
U (x, t + s)v(s)ds.
0
Then by taking ε = 2 and L > k in Lemma 3.10 we have
(3.13)
|Ũ (x, t)| ≤ C 0 exp k ||x|| + t , t ≥ 0.
Therefore, Ũ (x, t) is a continuous function of exponential type in
o
n
d+1
Rd+1
×
[0,
∞[=
(x,
t)
:
x
∈
R
,
t
≥
0
.
+
+
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
Title Page
Moreover, Ũ satisfies
(3.14)
Dunkl-Sobolev Spaces
(∂t − 4k,β )Ũ (x, t) = 0
in
Contents
Rd+1
+ ×]0, ∞[.
Hence if we set g(x) = Ũ (x, 0) then g is also a continuous function of exponential
type, so that g belongs to G∗0 .
Using (3.5) in Lemma 3.10, we obtain for t > 0
d
P (−4k,β )Ũ (x, t) = P −
Ũ (x, t)
dt
Z ∞
(3.15)
= U (x, t) +
U (x, t + s)w(s)ds.
0
R∞
If we set h(x) = − 0 U (x, s)w(s)ds then h is an entire function of exponential
type. As t → 0+ , (3.15) becomes
u = P (−4k,β )g(x) + h(x)
which completes the proof by replacing the coefficients an of P by (−1)n an .
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Theorem 3.12. Let U (x, t) be an infinitely differentiable function in Rd+1
+ ×]0, ∞[
satisfying the conditions:
i) (∂t − 4k,β )U (x, t) = 0 in Rd+1
+ ×]0, ∞[.
ii) There exist k > 0 and C > 0 such that
1
(3.16)
|U (x, t)| ≤ C exp k ||x|| +
, 0 < t < ε,
t
x ∈ Rd+1
+
for some ε > 0. Then there exists a unique element u ∈ G∗0 such that
U (x, t) = huy , Γk,β (t, x, y)i,
t>0
Dunkl-Sobolev Spaces
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
Title Page
and
lim+ U (x, t) = u
t→0
in G∗0 .
Proof. Consider a function, as in (3.12)
Z ∞
Ũ (x, t) =
U (x, t + s)v(s)ds.
0
U (x, t) = P (−4k,β )Ũ (x, t) + H(x, t),
where
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Then it follows from (3.13), (3.14) and (3.15) that Ũ (x, t) and H(x, t) are continuous
on Rd+1
+ × [0, ∞[ and
(3.17)
Contents
Z
H(x, t) = −
∞
U (x, t + s)w(s)ds.
0
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Furthermore, g(x) = Ũ (x, 0) and h(x) = H(x, 0) are continuous functions of exponential type on Rd+1
+ . Define u as
u = P (−4k,β )g(x) + h(x).
Then since P (−4k,β ) is a local operator, u belongs to G∗0 and
lim U (x, t) = u
t→0+
in
G∗0 .
Dunkl-Sobolev Spaces
Hatem Mejjaoli
Now define the generalized heat kernels for t > 0 as
Z
A(x, t) = (g ∗D,B pt )(x) =
g(y)Γk,β (t, x, y)dµk,β (y)
Rd+1
+
vol. 10, iss. 2, art. 55, 2009
Title Page
Contents
and
Z
B(x, t) = (h ∗D,B pt )(x) =
Rd+1
+
h(y)Γk,β (t, x, y)dµk,β (y).
Then it is easy to show that A(x, t) and B(x, t) converge locally uniformly to g(x)
and h(x) respectively so that they are continuous on Rd+1
+ × [0, ∞[, A(x, 0) = g(x),
and B(x, 0) = h(x). Now let V (x, t) = Ũ (x, t) − A(x, t) and W (x, t) = H(x, t) −
B(x, t). Then, since g and h are of exponential type, V (·, t) and W (·, t) are continuous functions of exponential type and V (x, 0) = 0, W (x, 0) = 0. Then by the
uniqueness theorem of the generalized heat equations we obtain that
Ũ (x, t) = g ∗D,B pt (x)
and
H(x, t) = h ∗D,B pt (x).
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It follows from these facts and (3.17) that
h
i
u ∗D,B pt = P (−4k,β )g + h ∗D,B pt
= P (−4k,β )Ũ (·, t) + H(·, t)
= U (·, t).
Now to prove the uniqueness of existence of such u ∈ G∗0 we assume that there exist
u, v ∈ G∗0 such that
U (x, t) = u ∗D,B pt (x) = v ∗D,B pt (x).
Then
Dunkl-Sobolev Spaces
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
Title Page
FD,B (u)FD,B (pt ) = FD,B (v)FD,B (pt )
which implies that FD,B (u) = FD,B (v), since FD,B (pt ) 6= 0. However, since the
Dunkl-Bessel transformation is an isomorphism we have u = v, which completes
the proof.
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4.
The Generalized Dunkl-Sobolev Spaces of Exponential Type
Definition 4.1. Let s be in R, 1 ≤ p < ∞. We define the space WGs,p
(Rd+1
+ ) by
∗ ,k,β
n
o
u ∈ G∗0 : es||ξ|| FD,B (u) ∈ Lpk,β (Rd+1
)
.
+
The norm on WGs,p
(Rd+1
+ ) is given by
∗ ,k,β
||u||WGs,p,k,β =
∗
mk,β
Dunkl-Sobolev Spaces
! p1
Z
Rd+1
+
eps||ξ|| |FD,B (u)(ξ)|p dµk,β (ξ)
Hatem Mejjaoli
.
vol. 10, iss. 2, art. 55, 2009
For p = 2 we provide this space with the scalar product
Z
(4.1)
hu, viW s,2 = mk,β
e2s||ξ|| FD,B (u)(ξ)FD,B (v)(ξ)dµk,β (ξ),
G∗ ,k,β
Title Page
Contents
Rd+1
+
and the norm
(4.2)
||u||2W s,2
G∗ ,k,β
= hu, uiW s,2
G∗ ,k,β
.
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Proposition 4.2.
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i) Let 1 ≤ p < +∞. The space
is a Banach space.
WGs,p
(Rd+1
+ )
∗ ,k,β
provided with the norm k·kW s,p
G∗ ,k,β
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ii) We have
2
d+1
WG0,2
(Rd+1
+ ) = Lk,β (R+ ).
∗ ,k,β
iii) Let 1 ≤ p < +∞ and s1 , s2 in R such that s1 ≥ s2 then
,p
s2 ,p
d+1
WGs∗1,k,β
(Rd+1
+ ) ,→ WG∗ ,k,β (R+ ).
ps||ξ||
Proof. i) It is clear that the space Lp (Rd+1
dµk,β (ξ)) is complete and since
+ ,e
0
FD,B is an isomorphism from G∗ onto itself, WGs,p
(Rd+1
+ ) is then a Banach space.
∗ ,k,β
The results ii) and iii) follow immediately from the definition of the generalized
Dunkl-Sobolev space of exponential type.
Proposition 4.3. Let 1 ≤ p < +∞, and s1 , s, s2 be three real numbers satisfying
s1 < s < s2 . Then, for all ε > 0, there exists a nonnegative constant Cε such that
for all u in WGs,p
(Rd+1
+ )
∗ ,k,β
Dunkl-Sobolev Spaces
Hatem Mejjaoli
(4.3)
,p
,p .
||u||WGs,p,k,β ≤ Cε ||u||WGs1 ,k,β
+ ε||u||WGs2 ,k,β
∗
∗
vol. 10, iss. 2, art. 55, 2009
∗
Proof. We consider s = (1 − t)s1 + ts2 , (with t ∈]0, 1[). Moreover it is easy to see
Title Page
t
s ,p .
||u||WGs,p,k,β ≤ ||u||1−t
s ,p ||u||
W 2
W 1
∗
G∗ ,k,β
Contents
G∗ ,k,β
Thus
t
,p
||u||WGs,p,k,β ≤ ε− 1−t ||u||WGs1 ,k,β
∗
∗
1−t ,p
ε||u||WGs2 ,k,β
t
t
Hence the proof is completed for Cε = ε
t
− 1−t
II
J
I
∗
,p
,p .
≤ ε− 1−t ||u||WGs1 ,k,β
+ ε||u||WGs2 ,k,β
∗
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Page 23 of 44
∗
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.
Proposition 4.4. For s in R, 1 ≤ p < ∞ and m in N, the operator 4m
k,β is continuous
s,p
s−ε,p
d+1
d+1
from WG∗ ,k,β (R+ ) into WG∗ ,k,β (R+ ) for any ε > 0.
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Proof. Let u be in WGs,p
(Rd+1
+ ), and m in N. From (3.1) we have
∗ ,k,β
Z
p
ep(s−ε)||ξ|| |FD,B (4m
k,β u)(ξ)| dµk,β (ξ)
Rd+1
+
Z
=
Rd+1
+
ep(s−ε)||ξ|| ||ξ||2mp |FD,B (u)(ξ)|p dµk,β (ξ).
As supξ∈Rd+1 ||ξ||2mp e−ε||ξ|| < ∞, for every m ∈ N and ε > 0
+
Z
p
ep(s−ε)||ξ|| |FD,B (4m
k,β u)(ξ)| dµk,β (ξ)
Dunkl-Sobolev Spaces
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
Rd+1
+
Z
≤
Rd+1
+
Title Page
eps||ξ|| |FD,B (u)(ξ)|p dµk,β (ξ) < +∞.
s−ε,p
d+1
Then 4m
k,β u belongs to WG∗ ,k,β (R+ ), and
s,p
||4m
k,β u||W s−ε,p ≤ ||u||WG ,k,β .
∗
G∗ ,k,β
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Page 24 of 44
1
Definition 4.5. We define the operator (−4k,β ) 2 by ∀x ∈ Rd+1
+ ,
Z
1
(−4k,β ) 2 u(x) = mk,β
Λ(x, ξ)||ξ||FD,B (u)(ξ)dµk,β (ξ), u ∈ S∗ (Rd+1 ).
Rd+1
+
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1
2
P
m
2
Proposition 4.6. Let P ((−4k,β ) ) = m∈N am (−4k,β ) be a fractional DunklBessel Laplace operators of infinite order such that there exist positive constants C
and r such that
rm
(4.4)
|am | ≤ C , for all m ∈ N.
m!
1
Let 1 ≤ p < +∞ and s in R. If an element u is in WGs,p
(Rd+1
+ ), then P ((−4k,β ) 2 )u
∗ ,k,β
s−r,p
d+1
belongs to WG∗ ,k,β (R+ ), and there exists a positive constant C such that
1
||P ((−4k,β ) 2 )u||W s−r,p ≤ C||u||WGs,p,k,β .
G∗ ,k,β
∗
Proof. The condition (4.4) gives that
|P (||ξ||)| ≤ C exp(r ||ξ||).
Dunkl-Sobolev Spaces
Thus the result is immediate.
Hatem Mejjaoli
1
2
Proposition 4.7. Let 1 ≤ p < +∞, t, s in R. The operator exp(t(−4k,β ) ) defined
by
vol. 10, iss. 2, art. 55, 2009
Title Page
1
2
∀x ∈ Rd+1
+ , exp(t(−4k,β ) )u(x)
Z
= mk,β
Rd+1
+
Λ(x, ξ)et||ξ|| FD,B (u)(ξ)dµk,β (ξ), u ∈ G∗ .
s−t,p
d+1
is an isomorphism from WGs,p
(Rd+1
+ ) onto WG∗ ,k,β (R+ ).
∗ ,k,β
Proof. For u in WGs,p
(Rd+1
+ ) it is easy to see that
∗ ,k,β
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1
k exp(t(−4k,β ) 2 )ukW s−t,p = ||u||WGs,p,k,β ,
G∗ ,k,β
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∗
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and thus we obtain the result.
−s,2
d+1
Proposition 4.8. The dual of WGs,2
(Rd+1
+ ) can be identified as WG∗ ,k,β (R+ ). The
∗ ,k,β
relation of the identification is given by
Z
(4.5)
hu, vik,β = mk,β
FD,B (u)(ξ)FD,B (v)(ξ)dµk,β (ξ),
Rd+1
+
−s,2
d+1
with u in WGs,2
(Rd+1
+ ) and v in WG∗ ,k,β (R+ ).
∗ ,k,β
Close
−s,2
d+1
Proof. Let u be in WGs,2
(Rd+1
+ ) and v in WG∗ ,k,β (R+ ). The Cauchy-Schwartz
∗ ,k,β
inequality gives
|hu, vik,β | ≤ ||u||W s,2 ||v||W −s,2 .
G∗ ,k,β
G∗ ,k,β
Thus for v in WG−s,2
(Rd+1
+ ) fixed we see that u 7−→ hu, vik,β is a continuous lin∗ ,k,β
s,2
d+1
ear form on WG∗ ,k,β (R+ ) whose norm does not exceed ||v||W −s,2 . Taking the
G∗ ,k,β
s,2
−1
d+1
−2s||ξ||
u0 = FD,B e
FD,B (v) element of WG∗ ,k,β (R+ ), we obtain hu0 , vik,β =
||v||W −s,2 .
G∗ ,k,β
Thus the norm of u 7−→ hu, vik,β is equal to ||v||W −s,2 , and we have then an
G∗ ,k,β
isometry:
s,2
d+1 0
d+1
)
.
(R
−→
W
WG−s,2
R
+
+
G
,k,β
,k,β
∗
∗
0
s,2
Conversely, let L be in WG∗ ,k,β (Rd+1
+ ) . By the Riesz representation theorem
s,2
d+1
and (4.1) there exists w in WG∗ ,k,β (R+ ) such that
L(u) = hu, wiW s,2
Z G∗ ,k,β
= mk,β
e2s||ξ|| FD,B (u)(ξ)FD,B (w)(ξ)dµk,β (ξ), for all u ∈ WGs,2
(Rd+1
+ ).
∗ ,k,β
Rd+1
+
If we set v = FD−1 e2s||ξ|| FD,B (w) then v belongs to WG−s,2
(Rd+1
+ ) and L(u) =
∗ ,k,β
s,2
d+1
hu, vik,β for all u in WG∗ ,k,β (R+ ), which completes the proof.
Proposition 4.9. Let s > 0. Then every u in WG−s,2
(Rd+1
+ ) can be represented as
∗ ,k,β
an infinite sum of fractional Dunkl-Bessel Laplace operators of square integrable
function g, in other words,
X sm
m
u=
(−4k,β ) 2 g.
m!
m∈N
Dunkl-Sobolev Spaces
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
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Proof. If u in WG−s,2
(Rd+1
+ ) then by definition
∗ ,k,β
e−s||ξ|| FD,B (u)(ξ) ∈ L2k,β (Rd+1
+ ),
which Proposition 2.1 ii) implies that
FD,B (u)(ξ)
FD,B (g)(ξ) = P
∈ L2k,β (Rd+1
+ ).
sm
m
||ξ||
m∈N m!
Dunkl-Sobolev Spaces
Hatem Mejjaoli
Hence, we have
vol. 10, iss. 2, art. 55, 2009
X sm
||ξ||m FD,B (g), in G∗0
m!
m∈N
X sm
m =
FD,B (−4k,β ) 2 g , in G∗0
m!
m∈N
FD,B (u) =
This completes the proof.
WGs,p
(Rd+1
+ )
∗ ,k,β
Proposition 4.10. Let 1 ≤ p < +∞. Every u in
function in the strip {z ∈ Cd+1 , || Im z|| < s} for s > 0.
is a holomorphic
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Proof. Let
Full Screen
Z
u(z) = mk,β
Rd+1
+
Λ(z, ξ)FD,B (u)(ξ)dµk,β (ξ),
z = x + iy.
From (2.4), for each µ in Nd+1 we have
µ
Dz Λ(z, ξ)FD,B (u)(ξ) ≤ ||ξ|||µ| e||y|| ||ξ|| |FD,B (u)(ξ)|.
Close
On the other hand from the Cauchy-Schwartz inequality we have
Z
||ξ µ ||e||y|| ||ξ|| |FD,B (u)(ξ)|dµk,β (ξ)
Rd+1
+
Z
≤
Rd+1
+
! 1q
||ξ||q|µ| eq||ξ||(||y||−s) dµk,β (ξ)
||u||WGs,p,k,β .
∗
Dunkl-Sobolev Spaces
Since the integral in the last part of the above inequality is integrable if ||y|| < s, the
result follows by the theorem of holomorphy under the integral sign.
Notations. Let m be in N. We denote by:
•
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
Title Page
0
Em
(Rd+1
+ )
the space of distributions on
less than or equal to m.
Rd+1
+
with compact support and order
d+1
0
0
• Eexp,m
(Rd+1
+ ) the space of distributions u in Em (R+ ) such that there exists a
positive constant C such that
|FD,B (u)(ξ)| ≤ Cem||ξ|| .
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Proposition 4.11.
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i) Let 1 ≤ p < +∞. For s in R such that s < −m, we have
0
Eexp,m
(Rd+1
+ )
⊂
WGs,p
(Rd+1
+ ).
∗ ,k,β
ii) Let 1 ≤ p < +∞. We have, for s < 0 and m an integer,
s,p
0
d+1
Em
(Rd+1
+ ) ⊂ WG∗ ,k,β (R+ ).
Close
Proof. The proof uses the same idea as Theorem 3.12 of [13].
Theorem 4.12. Let ψ be in G∗ . For all s in R, the mapping u 7→ ψu from WGs,2
(Rd+1
+ )
∗ ,k,β
into itself is continuous.
Proof. Firstly we assume that s > 0.
It is easy to see that
||v||2W s,2
G∗ ,k,β
≤
∞
X
(2s)j
j!
j=0
||v||2
j
2 (Rd+1 )
Ḣk,β
+
Dunkl-Sobolev Spaces
,
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
where k·kḢ s (Rd+1 ) designates the norm associated to the homogeneous Dunkl-Bessel+
k,β
Sobolev space defined by
Z
2
||v||Ḣ s (Rd+1 ) =
||ξ||2s |FD,B (v)(ξ)|2 dµk,β (ξ).
k,β
+
Rd+1
+
On the other hand, proceeding
as in Proposition 4.1 of [14] we prove that if u, v
T ∞ d+1
d+1
s
s
belong to Ḣk,β
(Rd+1
)
L
(R
+
+ ), s > 0 then uv ∈ Ḣk,β (R+ ) and
k,β
h
i
||uv||Ḣ s (Rd+1 ) ≤ C ||u||L∞ (Rd+1 ) ||v||Ḣ s (Rd+1 ) + ||v||L∞ (Rd+1 ) ||u||Ḣ s (Rd+1 ) .
k,β
+
k,β
+
k,β
Thus from this we deduce that for s > 0
∞
X
(2s)j
2
||ϕu||W s,2 ≤
||ϕu||2 j
2 (Rd+1 )
G∗ ,k,β
j!
Ḣk,β
+
j=0
h
≤ C ||ϕ||2L∞ (Rd+1 ) ||u||2W s,2
k,β
+
G∗ ,k,β
For s = 0 the result is immediate.
For s < 0 the result is obtained by duality.
+
k,β
+
k,β
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+
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+ ||u||2L∞
d+1
k,β (R+ )
||ϕ||2W s,2
G∗ ,k,β
i
< +∞.
5.
5.1.
Applications
Pseudo-differential-difference operators of exponential type
The pseudo-differential-difference operator A(x, 4k,β ) associated with the symbol
a(x, ξ) := A(x, −||ξ||2 ) is defined by
Z
Λ(x, ξ)a(x, ξ)FD,B (u)(ξ)dµk,β (ξ), u ∈ G∗ ,
(5.1) A(x, 4k,β )u (x) = mk,β
Rd+1
+
Hatem Mejjaoli
r
where a(x, ξ) belongs to the class Sexp
, r ≥ 0, defined below:
r
Definition 5.1. The function a(x, ξ) is said to be in Sexp
if and only if a(x, ξ) belongs
∞
d+1
d+1
to C (R
× R ) and for each compact set K ⊂ Rd+1
and each µ, ν in Nd+1 ,
+
there exists a constant CK = Cµ,ν,K such that the estimate
|Dξµ Dxν a(x, ξ)| ≤ CK exp(r ||ξ||),
Dunkl-Sobolev Spaces
(x, ξ) ∈ K × Rd+1
+
vol. 10, iss. 2, art. 55, 2009
Title Page
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hold true.
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Proposition 5.2. Let A(x, 4k,β ) be the pseudo-differential-difference operator asr
sociated with the symbol a(x, ξ) := A(x, −||ξ||2 ). If a(x, ξ) ∈ Sexp
then A(x, 4k,β )
d+1
in (5.1) is a well-defined mapping of G∗ into E∗ (R ).
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for all
Proof. For any compact set K ⊂ Rd+1
+ , we have
|a(x, ξ)| ≤ CK exp(r ||ξ||), for all (x, ξ) ∈ K × Rd+1
+ .
On the other hand since u is in G∗ the Cauchy-Schwartz inequality gives that
Z
|Λ(x, ξ)a(x, ξ)FD,B (u)(ξ)|dµk,β (ξ)
Rd+1
+
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Z
≤ CK ||u||W s,2
G,k,β
Rd+1
+
! 12
e−2(s−r)||ξ|| dµk,β (ξ)
is integrable for s > r. This prove the existence and the continuity of (A(x, 4k,β )u)(x)
for all x in Rd+1
+ . Finally the result follows by using Leibniz formula.
r,l
Now we consider the symbol which belongs to the class Sexp,rad
defined below:
Definition 5.3. Let r, l in R be real numbers with l > 0. The function a(x, ξ) is said
r,l
to be in Sexp,rad
if and only if a(x, ξ) is in C ∞ (Rd+1 × Rd+1 ), radial with respect to
the first d + 1 variables and for each L > 0, and for each µ, ν in Nd+1 , there exists
a constant C = Cr,µ,ν such that the estimate
Dunkl-Sobolev Spaces
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
Title Page
|Dξµ Dxν a(x, ξ)| ≤ CL|µ| µ! exp(r ||ξ|| − l ||x||)
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hold true.
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To obtain some deep and interesting results we need the following alternative
form of A(x, 4k,β ).
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Lemma 5.4. Let A(x, 4k,β ) be the pseudo-differential-difference operator associr,l
ated with the symbol a(x, ξ) := A(x, −||ξ||2 ). If a(x, ξ) is in Sexp,rad
then A(x, 4k,β )
in (5.1) is given by:
A(x, 4k,β )u (x)
#
"Z
Z
=mk,β
Λ(x, ξ)
τ−η FD,B (a)(·, η) (ξ)FD,B (u)(η)dµk,β (η) dµk,β (ξ)
Rd+1
+
Rd+1
+
for all u ∈ G∗ where all the involved integrals are absolutely convergent.
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Proof. When we proceed as [10] we see that for all L > 0 there exist C > 0 and
0 < τ < (Ld)−1 such that
(5.2)
|FD,B (a)(ξ, η)| ≤ C exp(r ||η|| − τ ||ξ||).
On the other hand since u in G∗ , we have
|FD,B (u)(η)| ≤ C exp(−t ||η||), ∀ t > 0.
Thus from the positivity for the Dunkl-Bessel translation operator for the radial functions we obtain
τ−η FD,B (a)(·, η) (ξ)FD,B (u)(η) ≤ C exp(−(t − r) ||η||)τ−η (exp(−τ k·k))(ξ).
Dunkl-Sobolev Spaces
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
Title Page
Moreover it is clear that the function η 7→ exp(−(t − r) ||η||)τ−η (exp(−τ k·k))(ξ)
belongs to L1k,β (Rd+1
+ ) for t > r, τ > 0. So that
Z
τ−η FD,B (a)(·, η) (ξ)FD,B (u)(η) dµk,β (η)
Rd+1
+
Z
≤C
Rd+1
+
exp(−(t − r) ||η||)τ−η (exp(−τ || · ||))(ξ)dµk,β (η).
The right-hand side is a Dunkl-Bessel convolution product of two integrable funcd+1
tions and hence is an integrable function on R+
. Therefore the function
Z
ξ 7→
τ−η (FD,B (a)(·, η))(ξ)FD,B (u)(η)dµk,β (η)
Rd+1
+
is in L1k,β (Rd+1
+ ). Applying the inverse Dunkl-Bessel transform we get the result.
Now we prove the fundamental result
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Theorem 5.5. Let A(x, 4k,β ) be the pseudo-differential-difference operator where
r,l
the symbol a(x, ξ) := A(x, −||ξ||2 ) belongs to Sexp,rad
. Then for all u in G∗ and all
s in R
||A(x, 4k,β )u||WGs,p,k,β ≤ Cs ||u||W r+s,p .
(5.3)
∗
G∗ ,k,β
Proof. We consider the function
Z
s||ξ||
Us (ξ) = e
τ−η FD,B (a)(·, η) (ξ)FD,B (u)(η)dµk,β (η),
Rd+1
+
Dunkl-Sobolev Spaces
s ∈ R.
Then invoking (5.2) and (2.19) we deduce that
Z
(5.4) |Us (ξ)| ≤
exp((r + s) ||η||)|FD,B (u)(η)|
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
Title Page
Contents
Rd+1
+
× τ−η exp(−(τ − |s|) ||y||) (ξ)dµk,β (η),
s ∈ R.
The integral of (5.4) can be considered as a Dunkl-Bessel convolution product between f (ξ) = exp(−(τ − |s|) ||ξ||) and g(ξ) = exp((r + s) ||ξ||)|FD,B (u)(ξ)|. It is
clear that f is radial and belongs to L1k,β (Rd+1
+ ) for τ > |s|, on the other hand since
p
d+1
FD,B (u) in G∗ then g in Lk,β (R+ ). Hence f ∗D,B g is in Lpk,β (Rd+1
+ ) and we have
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||f ∗D,B g||Lp
d+1
k,β (R+ )
≤ ||f ||L1
d+1
k,β (R+ )
||g||Lp
d+1
k,β (R+ )
= Cs ||u||W r+s,p .
G∗ ,k,β
Close
Thus
||A(x, 4k,β )u||WGs,p,k,β = ||Us ||Lp
∗
This proves (5.3).
d+1
k,β (R+ )
≤ ||f ∗D,B g||Lp
d+1
k,β (R+ )
≤ Cs ||u||W r+s,p .
G∗ ,k,β
5.2.
The Reproducing Kernels
Proposition 5.6. For s > 0, the Hilbert space WGs,2
(Rd+1
+ ) admits the reproducing
∗ ,k,β
kernel:
Z
Θk,β (x, y) = mk,β
Λ(x, ξ)Λ(−y, ξ)e−2s||ξ|| dµk,β (ξ),
Rd+1
+
that is:
Dunkl-Sobolev Spaces
s,2
d+1
i) For every y in Rd+1
+ , the function x 7→ Θk,β (x, y) belongs to WG∗ ,k,β (R+ ).
d+1
ii) For every f in WGs,2
(Rd+1
+ ) and y in R+ , we have
∗ ,k,β
hf, Θk,β (·, y)iW s,2
G∗ ,k,β
= f (y).
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
Title Page
Proof. i) Let y be in Rd+1
+ . From (2.5), the function
Contents
ξ 7→ Λ(−y, ξ)e−2s||ξ||
JJ
II
T 2
belongs to L1k,β (Rd+1
Lk,β (Rd+1
+ )
+ ) for s > 0, then from Proposition 2.1 ii) there
d+1
2
exists a function in Lk,β (R+ ), which we denote by Θk,β (·, y), such that
(5.5)
FD,B Θk,β (·, y) (ξ) = Λ(−y, ξ)e−2s||ξ|| .
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I
Thus Θk,β (x, y) is given by
Rd+1
+
Λ(x, ξ)Λ(−y, ξ)e−2s||ξ|| dµk,β (ξ).
d+1
ii) Let f be in WGs,2
(Rd+1
+ ) and y in R+ . From (4.1),(5.5) and (2.15) we have
∗ ,k,β
Z
hf, Θk,β (·, y)iW s,2 = mk,β
FD,B (f )(ξ)Λ(y, ξ)dµk,β (ξ) = f (y).
G∗ ,k,β
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Z
Θk,β (x, y) = mk,β
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Rd+1
+
Close
Proposition 5.7.
i) Let f be in G∗ . Then u(x, t) = Hk,β (t)f (x) solves the problem
(
∂
4k,β − ∂t
u(x, t) = 0 on Rd+1
+ ×]0, ∞[
u(·, 0) = f.
Dunkl-Sobolev Spaces
ii) The integral transform Hk,β (t), t > 0, is a bounded linear operator from
d+1
2
WGs,2
(Rd+1
+ ), s in R, into Lk,β (R+ ), and we have
∗ ,k,β
s2
kHk,β (t)f kL2
d+1
k,β (R+ )
≤ e 4t kf kW s,2
G∗ ,k,β
.
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
Title Page
Proof. i) This assertion follows from Definition 3.8 and Proposition 3.7 iii).
ii) Let f be in
WGs,2
(Rd+1
+ ).
∗ ,k,β
kHk,β (t)f k2L2
Using Proposition 2.1 ii) we have
d+1
k,β (R+ )
= mk,β kFD,B (Hk,β (t)f )k2L2
d+1
k,β (R+ )
.
Invoking the relation ships (2.21) and (3.2) we can write
Z
2
2
kHk,β (t)f kL2 (Rd+1 ) = mk,β
e−2t||ξ|| |FD,B (f )(ξ)|2 dµk,β (ξ).
k,β
+
Rd+1
+
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Therefore
kHk,β (t)f kL2
d+1
k,β (R+ )
s2
4t
≤ e kf kW s,2
G∗ ,k,β
.
Definition 5.8. Let r > 0, t ≥ 0 and s in R. We define the Hilbert space HGr,s∗ ,k,β (Rd+1
+ )
s,2
d+1
as the subspace of WG∗ ,k,β (R+ ) with the inner product:
hf, giHGr,s,k,β = rhf, giW s,2
∗
G∗ ,k,β
+hHk,β (t)f, Hk,β (t)giL2
d+1
k,β (R+ )
, f, g ∈ WGs,2
(Rd+1
+ ).
∗ ,k,β
The norm associated to the inner product is defined by:
kf k2H r,s
G∗ ,k,β
:= rkf k2W s,2
G∗ ,k,β
+ kHk,β (t)f k2L2
d+1
k,β (R+ )
.
HGr,s∗ ,k,β (Rd+1
+ )
Proposition 5.9. For s ∈ R, the Hilbert space
admits the following
reproducing kernel:
Z
Λ(x, ξ)Λ(−y, ξ)dµk,β (ξ)
Pr (x, y) = mk,β
.
re2s||ξ|| + e−2t||ξ||2
Rd+1
+
Rd+1
+ .
Proof. i) Let y be in
In the same way as in the proof of Proposition 5.6 i), we
can prove that the function x 7→ Pr (x, y) belongs to L2k,β (Rd+1
+ ) and we have
(5.6)
FD,B
Pr (·, y) (ξ) =
Λ(−y, ξ)
.
re2s||ξ|| + e−2t||ξ||2
Dunkl-Sobolev Spaces
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
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On the other hand we have
Close
(5.7) FD,B Hk,β (t)(Pr (·, y) (ξ)
= exp(−t||ξ||2 )FD,B Pr (·, y) (ξ),
for all ξ ∈ Rd+1
+ .
Hence from Proposition 2.1 ii), we obtain
Z
2 2
e−2t||ξ|| FD,B Pr (·, y) (ξ) 2 dµk,β (ξ)
kHk,β (t)(Pr (·, y)kL2 (Rd+1 ) = mk,β
k,β
+
Rd+1
+
≤
C
r2
2
Z
Rd+1
+
e−2t||ξ||
dµk,β (ξ) < ∞.
e4s||ξ||
Therefore we conclude that kPr (·, y)k2H r,s
< ∞.
Dunkl-Sobolev Spaces
G∗ ,k,β
Hatem Mejjaoli
d+1
ii) Let f be in HGr,s∗ ,k,β (Rd+1
+ ) and y in R+ . Then
(5.8)
vol. 10, iss. 2, art. 55, 2009
hf, Pr (·, y)iHGr,s,k,β = rI1 + I2 ,
∗
Title Page
where
I1 = hf, Pr (·, y)iW s,2
G∗ ,k,β
Contents
and
I2 = hHk,β (t)f, Hk,β (t)(Pr (·, y))iL2
d+1
k,β (R+ )
From (5.6) and (4.1), we have
Z
I1 = mk,β
Rd+1
+
.
e2s||ξ|| FD,B (f )(ξ)Λ(y, ξ)dµk,β (ξ)
.
re2s||ξ|| + e−2t||ξ||2
From (5.7) and (5.6) and Proposition 2.1 ii) we have
Z
2
e−2t||ξ|| FD,B (f )(ξ)Λ(y, ξ)dµk,β (ξ)
I2 = mk,β
.
re2s||ξ|| + e−2t||ξ||2
Rd+1
+
The relations (5.8) and (2.15) imply that
hf, Pr (·, y)iHGr,s,k,β = f (y).
∗
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5.3.
Extremal Function for the Generalized Heat Semigroup Transform
In this subsection, we prove for a given function g in L2k,β (Rd+1
+ ) that the infimum of
o
n
d+1
(R
)
rkf k2W s,2 + kg − Hk,β (t)f k2L2 (Rd+1 ) , f ∈ WGs,2
+
∗ ,k,β
G∗ ,k,β
k,β
+
∗
is attained at some unique function denoted by fr,g
, called the extremal function. We
start with the following fundamental theorem (cf. [11, 18]).
Theorem 5.10. Let HK be a Hilbert space admitting the reproducing kernel K(p, q)
on a set E and H a Hilbert space. Let L : HK → H be a bounded linear operator
on HK into H. For r > 0, introduce the inner product in HK and call it HKr as
hf1 , f2 iHKr = rhf1 , f2 iHK + hLf1 , Lf2 iH .
Dunkl-Sobolev Spaces
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
Title Page
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Then
i) HKr is the Hilbert space with the reproducing kernel Kr (p, q) on E which
satisfies the equation
K(·, q) = (rI + L∗ L)Kr (·, q),
where L∗ is the adjoint operator of L : HK → H.
ii) For any r > 0 and for any g in H, the infimum
n
o
inf rkf k2HK + kLf − gk2H
f ∈HK
∗
is attained by a unique function fr,g
in HK and this extremal function is given
by
(5.9)
∗
fr,g
(p) = hg, LKr (·, p)iH .
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We can now state the main result of this subsection.
Theorem 5.11. Let s ∈ R. For any g in L2k,β (Rd+1
+ ) and for any r > 0, the infimum
o
n
2
2
(5.10)
inf
rkf
k
+
kg
−
H
(t)f
k
k,β
L2 (Rd+1 )
W s,2
s,2
f ∈WG∗ ,k,β (Rd+1
+ )
G∗ ,k,β
k,β
+
∗
is attained by a unique function fr,g
given by
Z
∗
g(y)Qr (x, y)dµk,β (y),
(5.11)
fr,g (x) =
Dunkl-Sobolev Spaces
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
Rd+1
+
where
(5.12)
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Qr (x, y) = Qr,s (x, y)
Z
2
e−t||ξ|| Λ(x, ξ)Λ(−y, ξ)
= mk,β
dµk,β (ξ).
re2s||ξ|| + e−2t||ξ||2
Rd+1
+
Proof. By Proposition 5.9 and Theorem 5.10 ii), the infimum given by (5.10) is
∗
∗
attained by a unique function fr,g
, and from (5.9) the extremal function fr,g
is represented by
∗
fr,g
(y) = hg, Hk,β (t)(Pr (·, y))iL2 (Rd+1 ) , y ∈ Rd+1
+ ,
k,β
+
where Pr is the kernel given by Proposition 5.9. On the other hand we have
Hk,β (t)f (x)
Z
= mk,β
Rd+1
+
exp(−t||ξ||2 )FD,B (f )(ξ)Λ(x, ξ)dµk,β (ξ),
for all x ∈ Rd+1
+ .
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Hence by (5.6), we obtain
Z
Hk,β (t) Pr (·, y) (x) = mk,β
Rd+1
+
exp(−t||ξ||2 )Λ(x, ξ)Λ(−y, ξ)
dµk,β (ξ)
re2s||ξ|| + e−2t||ξ||2
= Qr (x, y).
This gives (5.12).
Corollary 5.12. Let s ∈ R, δ > 0 and g, gδ in L2k,β (Rd+1
+ ) such that
kg − gδ kL2
d+1
k,β (R+ )
≤ δ.
Dunkl-Sobolev Spaces
Hatem Mejjaoli
vol. 10, iss. 2, art. 55, 2009
Then
δ
≤ √ .
G∗ ,k,β
2 r
Proof. From (5.12) and Fubini’s theorem we have
∗
∗
kfr,g
− fr,g
k s,2
δ W
(5.13)
∗
FD,B (fr,g
)(ξ) =
Hence
−t||ξ||2
e
FD,B (g)(ξ)
.
2s||ξ||
re
+ e−2t||ξ||2
2
e−t||ξ|| FD,B (g − gδ )(ξ)
−
=
.
re2s||ξ|| + e−2t||ξ||2
Using the inequality (x + y)2 ≥ 4xy, we obtain
2
∗
∗
≤ 1 |FD,B (g − gδ )(ξ)|2 .
e2s||ξ|| FD,B (fr,g
− fr,g
)(ξ)
δ
4r
This and Proposition 2.1 ii) give
mk,β
∗
∗
kfr,g
− fr,g
k2 s,2 ≤
kFD,B (g − gδ )k2L2 (Rd+1 )
δ W
+
k,β
G∗ ,k,β
4r
1
≤ kg − gδ k2L2 (Rd+1 ) ,
+
k,β
4r
∗
FD,B (fr,g
∗
fr,g
)(ξ)
δ
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from which we obtain the desired result.
Corollary 5.13. Let s ∈ R. If f is in WGs,2
(Rd+1
+ ) and g = Hk,β (t)f . Then
∗ ,k,β
∗
kfr,g
− f k2W s,2
→0
as r → 0.
G∗ ,k,β
Proof. From (3.2), (5.13) we have
FD,B (f )(ξ) = exp(t||ξ||2 )FD,B (g)(ξ)
Dunkl-Sobolev Spaces
Hatem Mejjaoli
and
∗
FD,B (fr,g
)(ξ) =
−t||ξ||2
e
FD,B (g)(g)(ξ)
.
2s||ξ||
re
+ e−2t||ξ||2
vol. 10, iss. 2, art. 55, 2009
Title Page
Hence
∗
FD,B (fr,g
− f )(ξ) =
2s||ξ||
−re
FD,B (f )(ξ)
.
2s||ξ||
re
+ e−2t||ξ||2
Then we obtain
∗
kfr,g
− f k2W s,2
G∗ ,k,β
Z
=
Rd+1
+
with
hr,t,s (ξ) =
hr,t,s (ξ)|FD,B (f )(ξ)|2 dµk,β (ξ),
r2 e6s||ξ||
.
(re2s||ξ|| + e−2t||ξ||2 )2
Since
lim hr,t,s (ξ) = 0
r→0
and
|hr,t,s (ξ)| ≤ e2s||ξ|| ,
we obtain the result from the dominated convergence theorem.
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References
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hyperfunctions, Publ. RIMS, Kyoto Univ., 300 (1994), 203–208.
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Dunkl-Sobolev Spaces
Hatem Mejjaoli
[3] C. F. DUNKL, Differential-difference operators associated to reflection group,
Trans. Am. Math. Soc., 311 (1989), 167–183.
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Title Page
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[11] T. MATSUURA, S. SAITOH AND D.D. TRONG, Approximate and analytical
inversion formulas in heat conduction on multidimensional spaces, Journal of
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Dunkl-Sobolev Spaces
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